Now we can get back to our main task which starts with the field equations. They can be written as [1]}

mPl2(R-12gR)-2q2B=8T,      qmPlmG
.
The Bach term originates from the Weyl-squared term and is traceless. Since we also assume \(T^\mu _\mu =0\) , the trace of () is \(R=0\) . This can be taken to be one of the field equations and it can be written as

S”(x) S(x)-S'(x)2+1S(x)2=3 R”(x)R(x)
.
For the other equation we take the \(xx\)  component of (). For our choice of metric this takes a relatively simple form (with units \(m_{\rm Pl}=1\) ),

3 R'(x)2 S(x)2-2 R'(x) R(x) S'(x)S(x) -R(x)2R(x)2 S(x)2-q2S'(x)4+ S”(x)2S(x)2-2 S”'(x)S'(x) S(x)2-13 R(x)2 S(x)2
